M. Robert Showalter(1) and Stephen J. Kline(2)

When we derive a differential equation (defined at a point) from a coupled finite increment physical model, we must put ALL the variables and increments in our model equation into POINT FORM prior to algebraic simplification. The point forms of spatial quantities and time (in cm and second units) are:

length at a point: (1 cm)p             area at a point: (1 cm2)p

volume at a point: (1 cm3)p             a point in time: (1 second)p

with UNITS of length, area, volume, and time


NUMERICAL VALUES of    1 = length/length,     1= area/area

                                1 = volume/volume,   and         1= time/time respectively

We rederive the neural transmission equation using the point form of length, and find new terms. Some of these terms are enormous and dominant terms in the regime of neural function, and have important medical implications.


For reasons cited elsewhere(3), particularly Regan's data(4), we concluded that neural lines must conduct according to a law different from Kelvin-Rall (K-R)

with effective inductances ten or more orders of magnitude larger than K-R showed. We concluded that there had to be nonlinear terms, strongly dependent on diameter, that had been wrongly discarded. We quickly found terms in the derivation that appeared to be the right size and of the right form (on one interpretation). But these terms (on the conventional interpretation) were discarded as "infinitessimal." After a long process of elimination, we set out to question conventional limiting arguments used in physical mappings, not because we had substitute arguments in hand, but because we felt our evidence forced us to do so. We were not questioning formal mathematics in the axiomatic domain of the algebra. We were questioning procedures, that no one was really clear about, that mapped from physically measurable systems into the domain of the algebra. We were doing classical physics on a classical problem.

By far the hardest part of this investigation, for us and for people who have looked at our work, has been the idea that there COULD be a mistake in our procedures for going from measurable physical models to formal (axiomatic and abstract) models. From the start, we agreed with George Johnson's(5) admonition that:

" Scientists must constantly remind themselves that the map
is not the territory, that the models might not be capturing
the essence of the problem, and that the assumptions built
into a simulation might be wrong.(6) "

But for us and for others, established mapping patterns at the interface between measurable physics and mathematical representation seemed beyond doubt. It was difficult to question these patterns at all. Only after much time did it become clear to us that these usages, themselves, had to contain an error. Only after much more time did it occur to us that the error was at the level of arithmetic. Even so, we have found that our mapping usages between physical territory and analytical map do contain a mistake, and in particular an error in the usage of arithmetic. A rule restricting arithmetic with dimensional parameters has not been understood. As a consequence, invalid arithmetic has sometimes been accepted, and has sometimes misled us. No one with long experience and built-up reflexes of thought committed to the established mapping procedures can be asked to enjoy our conclusions. We can ask that they be evaluated on the basis of reason and evidence, even against the inclinations of reflex and emotion. We've found these conclusions difficult, too. At the same time, we believe that these conclusions, applied in neural science and medicine, will improve some lives, and save some lives. We also believe that these conclusions may facilitate other technical work.

We've had to reexamine derivational foundations that had worried J.C. Maxwell(7) (8). Can derivations be done using symbols that stand explicitly for the detailed physical quantities they represent, or is it necessary to abstract these symbols to numbers so that calculation can go on? We found that to answer this question, we needed a clearer understanding of the symbol constructs that we logically must use to represent "detailed physical quantities." We've found that there must be a physical, context-connected modelling domain that differs from the context-free domain of the algebra, that uses these symbol constructs. We've had to define the dimensional parameters, entities, long used without clear definition, that do not exist in the domain of the algebra. We then found that the dimensional parameters were subject to a special type restriction on multiplication and division, explained below, that did not correspond to anything in the domain of the algebra. Derivations of coupled physical models without knowledge of this type restriction can produce false infinitesimals, and sometimes false infinities. With the dimensional parameters defined and properly used, derivations CAN be done using symbols that are explicit representations of the detailed physical quantities they represent. These derivations can be converted, without misrepresentation, into abstract equations.

The dimensional parameters are the entities that interface between our experimental inferences and the formalities of abstract, symbolic mathematics. Our natural laws are not derived from axioms. They are logical inferences based on experiments.

All dimensional numbers, including the dimensional parameters, can be expressed in an explicit form as follows(9) (10):

Pn is a real number. The Xi's are dimensions that may be abstract dimensions, or that may be measurable dimensions such as length, charge, voltage, etc. The Xi's are raised to dimensional exponents. The value of Pn that represents a particular physical quantity depends on the units in which the dimensions are expressed(11) (12) (13). Units are conventional standards (such as those of the MKS or CGS system,) that measure dimensions or quantities expressed in terms of groups of dimensions.

The dimensional parameters are usually taken for granted, but we have found it necessary to point them out and define them. Here is an example of a dimensional parameter in use. The inductance, L, of a particular wire is the dimensional and numerical constant that expresses the physical relation between the gradient of voltage as a function of position, time and the derivative of current as a function of position and time. The inductance per unit length of a wire can be written

where v is volts, t is seconds, Q is coulombs, and x is cm. In the usual compact form, the units are implicit. We may write the natural law of inductance on the wire as follows:

We may set out this same relation with the numerical and dimensional parts explicitly written as dimensional numbers.

The numerical parts multiply, and the dimensional parts multiply in the sense of addition of exponents. Note that (4) and (5) are differential equations, defined at points even though both dv/dx and L include measurement notions set out per unit length. The unfamiliar notion of "length at a point" is inherent.

Here are some directly measurable dimensional parameters (often referred to as properties):

mass, density, viscosity, bulk modulus, thermal conductivity, thermal diffusivity, resistance (lumped), resistance (per unit length), inductance (lumped), inductance (per unit length), membrane current leakage (per length), capacitance (lumped), capacitance (per unit length), magnetic susceptibility, emittance, ionization potential, reluctance, resistivity, coefficient of restitution, and many more.

In addition to the directly measurable dimensional parameters and dimensional parametric functions there are also compound dimensional parameters, made up of the products or ratios of dimensional parameters and other dimensional variables. A famous class of the compound dimensional parameters is the dimensionless numbers, such as the Reynolds number(14).

DEFINITION: A dimensional parameter is a "dimensional transform ratio number" that relates two measurable functions numerically and dimensionally. The dimensional parameter is defined by measurements (or "hypothetical measurements") of two related measurable functions A and B. The dimensional parameter is the algebraically simplified expression of {A/B} as defined in A = {{A/B}} B. The dimensional parameter is a transform relation from one dimensional system to another. The dimensional parameter is also a numerical constant of proportionality between A and B (a parameter of the system.) The dimensional parameter is valid within specific domains of definition of A and B that are operationally defined by measurement procedures.

Example: A resistance per unit length determined for a specific wire for ONE specific length increment and ONE specific current works for an INFINITE SET of other length increments and currents on that wire (holding temperature the same.)

For example, for a length x along the wire, the voltage difference is set out

for this case, voltage drop is proportional to length between points of reference, x times (average) current over the interval.

The dimensional parameters are not axiomatic constructs. They are context-based linear constructs that encode experimental information. We've implicitly assumed that

"dimensional parameters are just dimensional numbers."

without any mathematical justification at all. This assumption makes the difference between the "first method" to the "second method" of Maxwell(15) (16).

For instance, derivations of a wire or line conduction model from a finite physical model yields equations(17) such as the following, which have many terms that would be rejected out of hand as "infinitesimal." For example, equation (7),

which may be "simplified" according to common arguments to (8)

Commonly, terms like those below the first line in (7 or 8) are dismissed out of hand as infinitesimal.    Difficulties with this dismissal on the basis of meaning have been set out elsewhere(18). Even setting these meaning issues aside, the dismissal fails several numerical closure tests. At a finite scale, before taking the limit, the terms below the first line are supposed to represent finite quantities. Yet when we take the limit as x goes to zero, these crossterms are infinitesimal, and are discarded in the differential equation. Now, let's take that differential equation, and integrate it back up to specific scale x. We get an equation that lacks the crossterms that we know existed at scale x in the first place. This is not closure. The derivation is inconsistent with itself, and the disparity can be numerically large.

The expressions in curly brackets in (7 and 8) also fail a conservation test that representations of physical circumstance must pass. Consider a curly bracketed expression in x2. If x is divided into ten pieces, and those ten subintervals are computed and summed, that sum is only 1/10 the value for the same expression computed over interval x, taken in one step. We can make the "physical" value on the interval x vary widely, depending on how many subintervals we choose to divide x into. This is not physical behavior.

These expressions are also numerically meaningless because they are constructed on the basis of a type invalid arithmetic, as shown below. A "number" or "expression" that can be manipulated by "proper arithmetic" and permissible unit changes so that it has any value at all is meaningless. Let's look at a simple loop test, analogous to many closure tests in physical logic. In Fig. 1, an algebraically unsimplified dimensional group that includes products or ratios of dimensional numbers, such as one of those in curly brackets in (7) or (8), is set out in cm length units at A. This quantity is algebraically simplified directly in cm units to produce "expression 1," dealing with the dimensional parameters and increments as "just numbers." The same physical quantity is also translated from A into a "per meter" basis at C. The translated quantity at C is then algebraically simplified to D in the same conventional fashion. The expression at D, expressed in meter length units, is converted to a "per cm" basis to produce "expression 2." Expression 1 and Expression 2 must be the same, or the calculation is not consistent with itself. Quantities like those in the curly brackets consistently fail the loop test. By choice of unit changes, and enough changes, such "quantities" can be made to have any numerical value at all. Expressions such as those in curly brackets in (7) or (8) are meaningless as usually interpreted.

The loop test fails!

The loop test fails because a standard procedure is flawed.

Before algebraic simplification, going from one unit system to another adjusts not just the numerical value of dimensional properties in the different unit systems, but numerical values corresponding to the spatial variable (length), as well.

After algebraic simplification, one has a compound dimensional property - adjusting it to a new unit system corresponds to adjusting numerical values that correspond to the unit change for the dimensional properties only, with no corresponding adjustment for the absorbed spatial variable.

The result is an irreversible, numerically absurd, but now standard mathematical operation. The difficulty has been deeply buried and hidden in our notation.

Note that the loop test of Fig. 1 only fails for terms that are now routinely discarded as "infinitesimal" or "infinite" without detailed examination.    Multiplication and division of groups of dimensional numbers (and dimensional parameters) without spatial or temporal increments works without difficulty, with numerical values handled arithmetically, and dimensional exponents added and subtracted in the usual way. Dimensional parameters associated with increments according to the exact pattern that defined the dimensional parameters also work.

Problems arise when we associate several dimensional parameters together with several increments, where the increments correspond to different physical effects evaluated over the same interval. The procedure we have used for evaluating such circumstances involves a kind of multiple counting that yields perverse results, as the loop test shows. We want to incorporate a rule that avoids mistakes like this. The rule needed restricts multiplication and division of dimensional parameters and increments to point form. It requires us to clarify our concept of increments (of length, area, volume, or time) defined at a point.

Notated as we have notated them, and interpreted as we have interpreted them, entities that represent coupled relations are physically unclear and numerically undefined. Although we may notate the interaction of resistance, capacitance, and resistance together in interaction with the measurable di/dt over the interval x as

where x is a number times a length unit, the loop test shows that this notation, literally interpreted, does not correspond to any consistent numerical value when unit systems are changed, and then changed back.

Let's rewrite (9) setting out a notation that makes explicit problems we need to solve concerning our notation of "length at a point" in this expression:

That is, we are trying to express "length" at a point (the asymptotic notion that a "differentially small" length reflects.) R and C are already in point form, both numerically and dimensionally. di/dt is defined at a point. If we had an point form for length, there would be no reason for a limiting argument in (9) or (10).

Let's think of what we already do when we reason from measurement. Our measurement procedures define things in terms of spatial variables (length, area, volume, time) and other dimensions (voltage, charge, and many others). The measurements are inherently finite in nature. Still, we speak of properties in POINT FORM, defined at points. (For instance, we speak of "resistance per unit length defined at a point" even though a point has 0 length. The numerical-scaling argument we use to arrive at point properties is simple and nearly reflexive. To intensify our properties, we say that

"the property at a point is the property that we get from a logic of interpolation from a finite scale to finer and finer scales. The interpolation assumes homogeneity of the model down to a vanishing spatial (and/or temporal) scale."

For example, consider the notion of resistance per unit length. Let's idealize the wire as a line. The resistance R expresses voltage gradient per unit length, per unit current. For any interval that includes length, the basic notion of resistance can be directly defined "per unit length". Other properties can be defined in similar ways "per unit area" or "per unit length" over finite areas, or finite volumes. But the notion of "length (or area, or volume) at a point" is an abstraction. This extremely useful and inescapable abstraction is much older(19)

than some of our rigorous calculus formality(20). In thermodynamics and elsewhere, we don't intensify our extensive variables by a calculus argument of any kind. We just assume that the property we're considering is homogeneous, and write our point form variables directly.

The abstract notion of length or area or volume "at a point" is already embedded in many of the point form properties in common use. Using cm and second units, the point forms of the properties of length, area, volume and time are:

Point form of length = { 1 length/length (length unit)}

length at a point in cm units: ( 1 cm )p

Point form of area = { 1 area/area ( (length unit)2 ) }

area at a point in cm units: ( 1 cm2 )p

Point form of volume = { 1 volume/volume ( (length unit)3) }

volume at a point in cm units: ( 1 cm3 )p

A point in time:

{ 1 (time increment)/(time increment) ( (time unit) ) }

a point in time in second time units: ( 1 second )p

The subscript is a marker, without arithmetic significance, that may be read as "at a point." The dimensions of length, area, and volume are length to the first, second and third power respectively. The numerical coefficients are identity operators, 1, because, for even the smallest imaginable numerical values of length, l, area, a; or volume v

l/l = 1 a/a = 1 v/v = 1 and t/t = 1

We can rewrite (10) as

Substituting the point form of length into (7) or (8) in place of x, we may algebraically simplify the bracketed expressions in the equation(s). This separates R, L, G and C into numerical parts (Rn, Ln, Cn, and Gn) that are algebraically simplified together, and unit groups that are algebraically simplified together (by adding exponents.) We'll choose a semi-arbitrary voltage-unit, charge-unit, cm, time-unit system here (v-Q-cm-t units.) We get:

The analogous di/dx equation is

Each term consists of one (compound) dimensional parameter times a measurable. These differential equations, when integrated to length x, reconstruct the values that apply to that length x, with no lost terms. Every term in these differential equations passes the loop test of Fig 1. We may map these differential equations symbol-for-symbol into corresponding partial differential equations. We may map these differential (or corresponding partial differential) equations symbol-for-symbol into the domain of the algebra. These equations are different equations from the Kelvin-Rall equations (1) now used in neurophysiology.

An important difference is the effective inductance term. For unmyelinated axons and dendrites in the neural range of sizes, the numerical magnitude of R2C/4 is between 1012 and 1021 times larger than L, depending on dendrite diameter and other variables. This term, which is much too small to measure in large scale electrical engineering(21), is a dominant and practically important term at neural scales in neural tissue.

Many terms now thought to be "infinities" are also finite terms when they are correctly interpreted in point form.

We have shown that physical domains, that include dimensional parameters that represent measurable circumstances, differ from the domain of the algebra. Unless we know this, we can discard important terms, and delude ourselves, or form false infinities, and delude ourselves. We can avoid this if we follow the following lesson:


When we represent a finite increment physical system in the form of a differential equation (defined at a point) we must put ALL the variables and increments into POINT FORM - it is not valid to have all the quantities except the increments in point form, with the increments in extensive form. The point forms of spatial quantities and time (expressed here in cm and second units) are:

length at a point: (1 cm)p          area at a point: (1 cm2)p

volume at a point: (1 cm3)p       a point in time: (1 second)p

with UNITS of length, area, volume, and time


NUMERICAL VALUES of 1 = length/length,     1= area/area

1 = volume/volume, and         1= time/time   respectively

With ALL the variables and increments in our equation representation set out in point form, algebraic simplification yields a differential equation that validly represents our system.


1. Department of Curriculum and Instruction, School of Education, University of Wisconsin, Madison, USA.

email: showalte@macc.wisc.edu

2. Department of Mechanical Engineering, Stanford University, Stanford Ca. USA.

3. Showalter, M.R. A (1997) Reasons to doubt the current neural conduction model. available FTP angus.macc.wisc.edu/pub2/showalt

4. David Regan (1989) HUMAN BRAIN ELECTROPHYSIOLOGY: Evoked Potentials and Evoked Magnetic Fields in Science and Medicine Elsevior pp. 106-108.

5. See http://www.santafe.edu/~johnson and Science Forums for THE NEW YORK TIMES at http://www.nytimes.com/.

6. Johnson, G. (1997) Proteins Outthink Computers in Giving Shape to Life NEW SCIENTIST March 27, 1997.


8. Showalter, M.R., Kline, S.J. A (1997) Modelling of physical systems according to Maxwell's first method. available FTP angus.macc.wisc.edu\pub2\showalt

9. Maxwell, op. cit.

10. 8. Kline, S.J. (1965, 1984) Similitude and Approximation Theory, McGraw-Hill; Springer-Verlag, Chapter 2.

11. Maxwell, op. cit.

12. 10. Kline op. cit. Chapter 2.

13. 11. Bridgman, P.W. (1922, 1931) Dimensional Analysis Yale University Press, New Haven, Chapters 2, 3.

14. Kline, op. cit. Chapter 3. See tables 3.1, 3.2.

15. Maxwell, op. cit.

16. Showalter & Kline A.

17. Showalter, M.R. B (1997) A new passive neural equation. Part a: derivation. available FTP angus.macc.wisc.edu\pub2\showalt

18. Showalter & Kline A.

19. In PRINCIPIA MATHEMATICA (1687) Book 2, following prop XL, Isaac Newton discusses the propagation of sound. He employs two numbers that moderns would call "dimensional parameters" in his treatment. The first is mass of air per unit volume at a point. The second is compressibility of air per unit volume at a point. These dimensional entities are only experimentally definable in finite terms, but they are set out in intensive (point) form. Numerically and dimensionally, the intensive and extensive form of these numbers is the same.

20. 18. Compare Newton in the 1680's versus the work of Weierstrass and his school in the 1870's, set out in H. Poincare L'oeuvre mathematique de Weierstrass" Acta Mathematica, XXII, 1989-1899, pp 1-18.

21. Showalter, M.R. and Kline, S.J. B (1997) If equations derived according to Maxwell's 1st method are right, inferences from experiment are only valid over a RESTRICTED range.

available FTP angus.macc.wisc.edu\pub2\showalt